Cohomogeneity one hypersurfaces in CP 2 and CH 2

Transcription

1 Cohomogeneity one hypersurfaces in CP 2 and CH 2 Cristina Vidal Castiñeira Universidade de Santiago de Compostela 1st September 2015 XXIV International Fall Workshop on Geometry and Physics, Zaragoza 2015 Supported by projects GRC , EM2014/009 and MTM P with FEDER funds (Spain)

2 Contents 1 Motivation 2 Examples in CP 2 and CH 2 3 Main Theorem 4 Applications

3 Motivation Problem Construct examples of hypersurfaces with certain geometric properties.

4 Motivation Problem Construct examples of hypersurfaces with certain geometric properties. M Riemannian manifold. H Lie group acting by isometries.

5 Motivation Problem Construct examples of hypersurfaces with certain geometric properties. M Riemannian manifold. H Lie group acting by isometries. We consider an isometric action H M M whose principal orbits have codimension 2. If we take a curve γ in M/H, the union of the corresponding principal orbits gives rise to a hypersurface M of M, which is a Riemannian manifold of cohomogeneity 1.

6 Motivation Problem Construct examples of hypersurfaces with certain geometric properties. M Riemannian manifold. H Lie group acting by isometries. We consider an isometric action H M M whose principal orbits have codimension 2. If we take a curve γ in M/H, the union of the corresponding principal orbits gives rise to a hypersurface M of M, which is a Riemannian manifold of cohomogeneity 1. M satisfies a property γ satisfies an ODE.

7 Motivation M Riemannian manifold

8 Motivation M Riemannian manifold H closed subgroup of Isom( M)

9 Motivation M Riemannian manifold H closed subgroup of Isom( M) H M M, (g, p) g(p) isometric action

10 Motivation M Riemannian manifold H closed subgroup of Isom( M) H M M, (g, p) g(p) isometric action Definition The codimension of the maximal dimensional orbits is called cohomogeneity of the action.

11 Motivation M Riemannian manifold H closed subgroup of Isom( M) H M M, (g, p) g(p) isometric action Definition The codimension of the maximal dimensional orbits is called cohomogeneity of the action. H M M is of cohomogeneity 0 M is a homogeneous Riemannian manifold.

12 Motivation M Riemannian manifold H closed subgroup of Isom( M) H M M, (g, p) g(p) isometric action Definition The codimension of the maximal dimensional orbits is called cohomogeneity of the action. H M M is of cohomogeneity 0 M is a homogeneous Riemannian manifold. H M M is cohomogeneity 1 the action has codimension one orbits, which are extrinsically homogeneous hypersurfaces.

13 Motivation H subgroup of Isom( M). Definition An isometric action of H on M is polar if there exists a submanifold Σ that intersects all the orbits of H orthogonally. The submanifold Σ is called a section.

14 Motivation H subgroup of Isom( M). Definition An isometric action of H on M is polar if there exists a submanifold Σ that intersects all the orbits of H orthogonally. The submanifold Σ is called a section.

15 Motivation H subgroup of Isom( M). Definition An isometric action of H on M is polar if there exists a submanifold Σ that intersects all the orbits of H orthogonally. The submanifold Σ is called a section. We consider a polar action H M M whose principal orbits have codimension 2. If we take a curve γ in the regular part of Σ, M = H γ = {h(γ(t)) : h H, t ( ε, ε)} is a Riemannian manifold of cohomogeneity 1.

16 Basic concepts and terminology M 2 (c) {CP 2, CH 2 } M real hypersurface ξ unit normal vector field on M Shape operator: SX = X ξ Principal curvatures: eigenvalues of S

17 Basic concepts and terminology M 2 (c) {CP 2, CH 2 } M real hypersurface ξ unit normal vector field on M Shape operator: SX = X ξ Principal curvatures: eigenvalues of S Definition M is a Hopf hypersurface Jξ is a principal curvature vector field.

18 Basic concepts and terminology M 2 (c) {CP 2, CH 2 } M real hypersurface ξ unit normal vector field on M Shape operator: SX = X ξ Principal curvatures: eigenvalues of S Definition M is a Hopf hypersurface Jξ is a principal curvature vector field. Definition h number of projections of Jξ into the principal curvature spaces. h = 1 M is Hopf.

19 Examples of Hopf and h = 2 hypersurfaces in CP 2 and CH 2 In CP 2, Wang (1973). Every homogeneous hypersurface is an open part of: A geodesic sphere (which is also a tube around a totally geodesic complex projective line CP 1 ) Tube around a totally geodesic real projective plane RP 2

20 Examples of Hopf and h = 2 hypersurfaces in CP 2 and CH 2 In CP 2, Wang (1973). Every homogeneous hypersurface is an open part of: A geodesic sphere (which is also a tube around a totally geodesic complex projective line CP 1 ) Tube around a totally geodesic real projective plane RP 2 In CH 2, J. Berndt and J. C. Díaz-Ramos (2007). Every homogeneous hypersurface is an open part of: A geodesic sphere Tube around a totally geodesic complex hyperbolic line CH 1 Tube around a totally geodesic real hyperbolic plane RH 2 A horosphere The ruled real hypersurface W 3 or one of its equidistant hypersurfaces.

21 Explain W 3 p CH 2, x CH 2 ( ) CH 2 p x

22 Explain W 3 p CH 2, x CH 2 ( ) CH 2 CH 1 p x

23 Explain W 3 p CH 2, x CH 2 ( ) CH 2 Ξ CH 1 p x

24 Explain W 3 p CH 2, x CH 2 ( ) CH 2 horocycle Ξ CH 1 p x

25 Explain W 3 p CH 2, x CH 2 ( ) CH 2 horocycle Ξ CH 1 p x

26 Strongly 2-Hopf hypersurfaces M real hypersurface in CP 2 and CH 2 ξ unit normal vector field on M S shape operator Definition (Ivey and Ryan) M is 2-Hopf if: The smallest S-invariant distribution D of M that contains Jξ has rank 2, D is integrable. Moreover, if the spectrum of S D is constant along the integral submanifolds of D, M is called strongly 2-Hopf.

27 Main Theorem Main Theorem A real hypersurface M in M 2 (c) {CP 2, CH 2 } is strongly 2-Hopf if and only if it is an open part of an everywhere non-hopf hypersurface H γ, where H connected subgroup of Isom( M 2 (c)) acting polarly on M 2 (c) with section Σ. γ : I Σ reg is a curve in the regular part of Σ.

28 Applications Local construction of hypersurfaces with 2 nonconstant principal curvatures in CP 2 and CH 2. [J.C. Díaz-Ramos, M. Domínguez-Vázquez, C. Vidal-Castiñeira: Real hypersurfaces with two principal curvatures in complex projective and hyperbolic planes, arxiv: v1]

29 Applications Question (Niebergall and Ryan) Are there real hypersurfaces in CP 2 and CH 2 with 2 principal curvatures, other than the standard examples? In a joint work with J.C. Díaz-Ramos and M. Domínguez-Vázquez we answered this question.

30 Applications H = U(1) U(1) U(1) acts on C 3. Take the quotient by the Hopf map (z λz, with λ S 1, z C 3 ). H acts polarly on CP 2 with section Σ = RP 2 and with principal orbits S 1 S 1 [Podestà, Thorbergsson].

31 Applications H = U(1) U(1) U(1) acts on C 3. Take the quotient by the Hopf map (z λz, with λ S 1, z C 3 ). H acts polarly on CP 2 with section Σ = RP 2 and with principal orbits S 1 S 1 [Podestà, Thorbergsson].

32 Applications H = U(1) U(1) U(1) acts on C 3. Take the quotient by the Hopf map (z λz, with λ S 1, z C 3 ). H acts polarly on CP 2 with section Σ = RP 2 and with principal orbits S 1 S 1 [Podestà, Thorbergsson]. We look for a curve γ in Σ such that H γ = {h(γ(t)) : h H, t ( ε, ε)} has two (generically nonconstant) principal curvatures. This is equivalent to the fact that γ satisfy certain ODE.

33 Applications Main Theorem Let H = U(1) U(1) U(1) act polarly on CP 2 with section Σ = RP 2. Then, for each regular p Σ and each tangent vector v T p Σ, there are exactly two curves γ i : ( ε, ε) Σ, i = 1, 2, with γ i (0) = p and γ i (0) = v, such that the set H γ i = {h(γ i (t)) : h H, t ( ε, ε)} is a real hypersurface with two (generically nonconstant) principal curvatures. Conversely, each non-hopf real hypersurface of CP 2 with two nonconstant principal curvatures is locally congruent to a hypersurface constructed as above.

34 Applications Local construction of hypersurfaces with 2 nonconstant principal curvatures in CP 2 and CH 2. [J.C. Díaz-Ramos, M. Domínguez-Vázquez, C. Vidal-Castiñeira: Real hypersurfaces with two principal curvatures in complex projective and hyperbolic planes, arxiv: v1] Local construction of constant mean curvature hypersurfaces of cohomogeneity one on M 2 (c). [C. Gorodski, N. Gusevskii: Complete minimal hypersurfaces in complex hyperbolic space, Manuscripta Math. 103 (2000) ]

35 Applications Local construction of hypersurfaces with 2 nonconstant principal curvatures in CP 2 and CH 2. [J.C. Díaz-Ramos, M. Domínguez-Vázquez, C. Vidal-Castiñeira: Real hypersurfaces with two principal curvatures in complex projective and hyperbolic planes, arxiv: v1] Local construction of constant mean curvature hypersurfaces of cohomogeneity one on M 2 (c). [C. Gorodski, N. Gusevskii: Complete minimal hypersurfaces in complex hyperbolic space, Manuscripta Math. 103 (2000) ] Examples of Levi-flat hypersurfaces in M 2 (c). [Y.-T. Siu: Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension 3, Ann. of Math. 151 (2000) ]

36 Applications T p M tangent space to M at p. ν p M normal space to M at p. N p maximal holomorphic subspace of T p M. II second fundamental form of M. Definition (Levi-flat) The Levi form of M at p is a symmetric bilinear form L : N p N p ν p M, such that for all u, v N p, all p M. M is Levi-flat L = 0. L(u, v) = II (u, v) + II (Ju, Jv),

37 Applications Local construction of hypersurfaces with 2 nonconstant principal curvatures in CP 2 and CH 2. [J.C. Díaz-Ramos, M. Domínguez-Vázquez, C. Vidal-Castiñeira: Real hypersurfaces with two principal curvatures in complex projective and hyperbolic planes, arxiv: v1] Local construction of constant mean curvature hypersurfaces of cohomogeneity one on M 2 (c). [C. Gorodski, N. Gusevskii: Complete minimal hypersurfaces in complex hyperbolic space, Manuscripta Math. 103 (2000) ] Examples of Levi-flat hypersurfaces in M 2 (c). [Y.-T. Siu: Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension 3, Ann. of Math. 151 (2000) ] An austere hypersurface in M 2 (c) with h 2 with 3 principal curvatures ± c 2, 0 at some point must be an open part of a W 3 in CH 2.